TST2025 #0l5z — TST2025 · USAMO Guide - Olympiad Mathematics Learning Platform

TST2025 #0l5z

TST2025Hard

Community difficultyNo ratings yetBe the first to rate this problem.

Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Problem

Let $A_1A_2\cdots A_{2025}$ be a convex 2025-gon, and let $A_i = A_{i+2025}$ for all integers $i$ . Distinct points $P$ and $Q$ lie in its interior such that $\angle A_{i-1}A_iP = \angle QA_iA_{i+1}$ for all $i$ . Define points $P_i^j$ and $Q_i^j$ for integers $i$ and positive integers $j$ as follows:

For all $i$ , $P_i^1 = Q_i^1 = A_i$ .
For all $i$ and $j$ , $P_i^{j+1}$ and $Q_i^{j+1}$ are the circumcenters of $PP_i^jP_{i+1}^j$ and $QQ_i^jQ_{i+1}^j$ , respectively. Let $\mathcal{P}$ and $\mathcal{Q}$ be the polygons $P_1^{2025}P_2^{2025}\cdots P_{2025}^{2025}$ and $Q_1^{2025}Q_2^{2025}\cdots Q_{2025}^{2025}$ , respectively. (a) Prove that $\mathcal{P}$ and $\mathcal{Q}$ are cyclic. (b) Let $O_P$ and $O_Q$ be the circumcenters of $\mathcal{P}$ and $\mathcal{Q}$ , respectively. Assuming that $O_P \neq O_Q$ , show that $O_P O_Q$ is parallel to $PQ$ .